IF you already have micromesh--- fresh micromesh, not stuff that has one foot in the grave--- use micromesh followed by liquid Brasso. I wet-sand to 600 before getting out the MM. I typically do every grit of micromesh for 15-30 seconds, with LIGHT pressure using water from a cheap ketchup squirt bottle from Walmart. Acrylic takes more work to get to a glassy shine than wood because it shows any and every imperfection/scratch.
Follow up with Brasso on a paper towel, and spend about 30 seconds on each blank with the lathe at HIGH speed (3700+ rmp), but do NOT build up extreme heat. You will get some heat, but don't get the blank "hot". Keep the paper towel wet with Brasso.
HOWEVER----------------------------
If you do not have micromesh, save your money and time just go to HF, Woodcraft, Klingspor, or wherever, and buy TWO buffing wheels-- at least 4" diameter, I use 8", and a mandrel to hold them--- I use mandrels that I hold in a drill chuck on my lathe. Also buy one brick of Tripoli, one brick of White Diamond (both are cheap).
Now, follow the instructions of this tutorial, found at the bottom, for buffing. I PROMISE the gloss is perfect and literally half the time or even a third of the micromesh deal. You can, if you want a stupid insane gloss, follow all this with Brasso. But it's really not necessary, just kinda fun.
http://www.thepenshop.net/cigar.pdf